J. Phys. Chem. A 2005, 109, 10997-11005
10997
Quantum Chemical Investigation of Hyperfine Coupling Constants on First Coordination Sphere Water Molecule of Gadolinium(III) Aqua Complexes Oleg V. Yazyev,† Lothar Helm,*,† Vladimir G. Malkin,‡ and Olga L. Malkina‡ Laboratoire de Chimie Inorganique et Bioinorganique, Ecole Polytechnique Fe´ de´ rale de Lausanne, EPFL-BCH, CH-1015 Lausanne, Switzerland, and Institute of Inorganic Chemistry, SloVak Academy of Sciences, SK-84536 BratislaVa, SloVak Republic ReceiVed: July 12, 2005; In Final Form: September 21, 2005
Hyperfine interactions (HFI) on the nuclei of the first coordination sphere water molecules in a model [Gd(H2O)8]3+ aqua complex and in the magnetic resonance imaging contrast agent [Gd(DOTA)(H2O)]- were studied theoretically. Density functional theory (DFT) calculations combined with classical molecular dynamics (MD) simulations have been used in order to take into account dynamic effects in aqueous solution. DFT relativistic calculations show a strong spin-polarization of the first coordination sphere water molecules. This spin-polarization leads to a positive 17O isotropic hyperfine coupling constant (Aiso(17O) ) 0.58 ( 0.11 MHz) and to a significant increase of the effective distance (〈reff(Gd-O)〉 ) 2.72 ( 0.06 Å) of dipolar interaction compared to the mean internuclear distance (〈r(Gd-O)〉 ) 2.56 ( 0.06 Å) obtained from the MD trajectory of [Gd(DOTA)(H2O)]- in aqueous solution. The point-dipole model for anisotropic hyperfine interaction overestimates therefore the longitudinal relaxation rate of the 17O nucleus by ∼45%. The 1H isotropic hyperfine coupling constant of the bound water molecule is predicted to be very small (Aiso(1H) ) 0.03 ( 0.02 MHz), and the point-dipole approximation for first coordination sphere water protons holds. The calculated hyperfine parameters are in good agreement with available experimental data.
Introduction Hyperfine interactions (HFI), the interactions between nuclear spin and electron spin in paramagnetic systems, are of great interest for investigation of structure and dynamics in bioinorganic chemistry. The common experimental techniques that permit direct measurement of HFI are electron spin resonance (ESR), nuclear magnetic resonance (NMR), and paramagnetic relaxation enhancement (PRE) of paramagnetic systems. In the systems involving paramagnetic d-transition metal and lanthanide ions one usually separates for practical reasons the metal HFI from HFI of distant (ligand) nuclei, which are also referred to as superhyperfine interactions. A detailed study of ligand HFI in paramagnetic systems can provide researchers with a considerable amount of structural information. As examples of the importance of the ligand HFI one can mention its decisive role in the description of NMR relaxation of solvent nuclei in solutions of paramagnetic species1 and its impact on rational development of novel contrast agents for magnetic resonance imaging (MRI) for medical applications.2-6 Currently, typical representatives of MRI contrast agents are gadolinium(III) complexes because of their high spin state (8S state, half-filled f-shell) and slow electron spin relaxation. Gd(III) based MRI contrast agents are probably one of the most striking examples where the purely quantum nature of HFI directly leads to a diagnostic tool, namely, increase in contrast in an MRI image, which cannot be described by means of any kind of classical theory. * Corresponding author. Mailing address: Laboratoire de Chimie Inorganique et Bioinorganique, Ecole Polytechnique Fe´de´rale de Lausanne, EPFL-BCH; CH-1015 Lausanne, Switzerland. Tel: +41 21 693 9876. Fax: +41 21 693 9875. E-mail:
[email protected]. † Ecole Polytechnique Fe ´ de´rale de Lausanne. ‡ Slovak Academy of Sciences.
Furthermore, paramagnetic d-transition metal and lanthanide ions are used as natural and artificial probes to study structure of biological objects using experimental techniques mentioned above.7 For instance, PRE studies of systems containing highly paramagnetic Mn2+ and Gd3+ ions can provide ultimate longrange structural constraints up to 40 Å for determination of the structure of biological objects.1,8-11 Electron-nuclear double resonance (ENDOR) and state-of-the-art electron spin-echo envelope modulation (ESEEM) methods can locate individual nuclei in situations where it could not be done using other methods.12,13 It is well-known from experiment that the main contribution to the paramagnetic enhancement of longitudinal relaxation comes from dipolar contribution to HFI. The PRE for first coordination sphere water molecules is commonly described by Solomon-Bloembergen-Morgan (SBM) equations.14-17 In the original formulation, these equations suggest the point-dipole approximation in which unpaired electrons are considered as localized at the position of the paramagnetic metal center. Any spin distribution effects over the system are therefore neglected. Due to this approximation the dipolar contribution to HFI can be described by using only information about the distance between the nucleus of interest and the nucleus of the paramagnetic center, information which can be obtained from diffraction experiments, for example. It is also worth mentioning that full knowledge of the dipolar HFI tensor is not easily accessible with existing experimental techniques and therefore the point-dipole approximation is inevitable. Quantum chemical calculations can provide some a priori knowledge of dipolar HFI. The first step in this direction was made by Kowalewski et al.18-20 in early studies of model aqua complexes of first row transition metal ions using the unrestricted Hartree-Fock (UHF) method. Their results clearly
10.1021/jp053825+ CCC: $30.25 © 2005 American Chemical Society Published on Web 11/10/2005
10998 J. Phys. Chem. A, Vol. 109, No. 48, 2005 demonstrated the limitations of the point-dipole approximation. Later UHF studies of Das et al.21-23 and density functional theory (DFT) calculations of Clostridium pasteurianum iron(III) rubredoxin protein model system by Wilkens et al.24 confirmed this conclusion. Quantum chemical description of the system makes it possible to generalize the Solomon-Bloembergen-Morgan equations and to avoid the point-dipole approximation even at the estimative level. The origin of ligand HFI in complexes containing a paramagnetic lanthanide ion is different from that of d-transition metal ions. Due to the core character of the f-shell any significant contribution of ligand atomic orbitals to singly occupied molecular orbitals (SOMOs) is improbable.25 However, the effect called spin-polarization26 may manifest in lanthanide complexes. Early studies27 confirmed the reliability of the pointdipole approximation for spherically symmetric f 7 ions, but at that time it was not possible to apply the whole range of quantum chemistry methods available today. A more recent quantum chemical study28 already clarified the possibility of spin-polarization effect in Gd3+-H2O complexes. Nevertheless, a more detailed knowledge about the mechanisms of HFI in lanthanide complexes is required to obtain reliable parameters for structural and dynamic investigations. In the past decade an impetuous progress in the development of quantum chemical approaches for studying HFI took place. The density functional theory (DFT) technique can now be routinely applied for investigation of HFI in model systems of hundreds of atoms in size with an accuracy approaching the experimental one.26,29 In the present work we study in detail both isotropic and dipolar HFI on 1H and 17O of a first coordination sphere water molecule in Gd(III) complexes. The work is composed of two parts. The first, methodological part serves for the validation of DFT as an accurate tool for description of HFI in lanthanide complexes where the amount of reliable experimental data concerning hyperfine coupling constants is limited. Keeping in mind that relativistic effects play an important role for heavy element compounds we use state-of-the-art relativistic theory all-electron approaches. A simple symmetric system modeling the Gd3+ octaaquaion is used for this first part. The second part considers the HFI on the first coordination sphere water molecule of the [Gd(DOTA)(H2O)]- complex. This chelate complex is a typical member of the family of Gd3+ complexes which are actually used in medical applications as MRI contrast agents. To take into account the dynamic nature of the complex in aqueous solution as well as to investigate the influence of geometric parameters such as the ion-water distance on HFI, we use an approach which combines classical MD simulations with quantum chemistry calculations of HFI. The main purpose of this work is to show the limitations of point-dipole approximation for dipolar contributions to HFI of first coordination sphere water molecules in Gd(III) complexes and to propose corrections if possible. The deep understanding of underlying physical phenomena in this class of chemical compounds is a necessary element for future development of effective MRI contrast agents for medical and other applications. The outline of the article is as follows: First we review the definitions and models of hyperfine interactions and nuclear relaxation in the section Theoretical Foundation. Then the choice of model systems and details of quantum chemistry methods are explained in the section Computational Techniques. In the next section, Results and Discussion, the performance of the applied computational methods is discussed. The calculated ligand hyperfine parameters and the phenomenon of spin-
Yazyev et al. polarization for model Gd3+ octaaqua complex and more realistic [Gd(DOTA)(H2O)]- in solution are also considered. The predicted constants are compared with the existing experimental data. In the last subsection, the conclusions for evaluation of relaxation data are made. Theoretical Foundation Hyperfine interactions are magnetic interactions between nuclear and electron spins. The corresponding spin-Hamiltonian of this interaction can be written as
H ) S‚A‚I
(1)
where A is the 3 × 3 HFI tensor and S and I are the vectors of the electron and the nuclear spin, respectively. The HFI tensor can be split into isotropic and anisotropic parts,
A ) Aiso1 + T
(2)
where Aiso is the isotropic hyperfine coupling constant (scalar); 1 is the 3 × 3 unit matrix, and T is the traceless matrix of the anisotropic contribution. Within the commonly used Breit-Pauli approximation,30 the scalar isotropic (Fermi contact) hyperfine coupling constant on nucleus N is
Aiso(N) )
4π β β g g FR-β(RΝ) 3S e N e N
(3)
and a matrix element of the anisotropic (dipolar) contribution is
Tij(N) )
∫
1 β β g g FR-β(r) × 2S e N e N (r - RN)2δij - 3(ri - RNi)‚(rj - RNj) dr (4) (r - RN)5
where βe and βN are Bohr and nuclear magnetons, respectively, ge and gN are free-electron and nuclear g-values, and S is the total electronic spin of the atom, ion, or molecule. Thus, both contributions depend only on the distribution of the electron spin density, the difference between majority spin (R) and minority spin (β) densities, FR-β(r) ) FR(r) - Fβ(r), of the system in the spin state S and physical constants. The physical interpretations of the two contributions to A are as follows. The isotropic contribution (Fermi contact), Aiso, is proportional to the value of the spin density at the position of nucleus N and thus it possesses a local character. Physically this contribution represents a magnetic field generated at the point of nucleus by the presence of the electron magnetic moment itself. On the contrary, the anisotropic contribution, T, is the dipolar integral over the whole space and has therefore a nonlocal character. It represents the dipole-dipole type of magnetic interaction between the magnetic moments of nuclear and electron spins. The dipolar contribution vanishes if the spin density is highly symmetric when observed from the point of nucleus N. In many cases the spin density distribution is determined by the shapes of singly occupied molecular orbitals (SOMOs). This contribution is usually referred to as spin-delocalization and it is always positive, if one follows the above-mentioned convention for R- and β-electron densities. A second part of spin density comes from so-called spin-polarization effects, which originate from nonequal potentials experienced by R and β electrons and orthogonality constraints imposed on MOs: this leads to different shapes of R and β MOs (which otherwise would be doubly occupied). This contribution can be positive as well as
Hyperfine Interactions in Gd(III) Complexes
J. Phys. Chem. A, Vol. 109, No. 48, 2005 10999
negative at the location of nucleus N, but it always integrates to zero. Spin-polarization effects can be described in short as “an effective attraction”: the localized unpaired electrons “attract” the nearby ones of the same spin. This can result in a negative spin density in the vicinity of SOMO nodes and where SOMO density is vanishing and in a slight increase of positive spin density, produced by the spin-delocalization effect. Spinpolarization is often referred to as a second-order effect.31 The superposition of these two effects will be referred to as spindistribution from now on. In chemical systems involving paramagnetic metal ions it is common to separate the notions of hyperfine coupling constants on the metal nucleus and on ligand nuclei. In this work we discuss only ligand hyperfine interactions which are often referred to as superhyperfine coupling. The simplest possible model for ligand HFI is the point-dipole approximation.20 Within this approximation spin-distribution effects are neglected and the dipolar ligand HFI tensor TPD depends only on the distance between the metal and ligand nuclei, rMX. Thus, TPD can be written in form32
TPD ) βeβNgegN
1 rMX3
(
2 0 0 0 -1 0 0 0 -1
)
(5)
assuming that the metal-ligand nucleus vector is (rMX, 0, 0). It is also worth mentioning that this approximation results in a zero isotropic hyperfine coupling constant on all ligand nuclei since all the spin density is located on the metal ion. Thus, nonzero ligand isotropic hyperfine coupling constants (which are often experimentally measurable) tell us immediately about the deficiency of the point-dipole approximation. The sign of the ligand hyperfine coupling constant provides the sign of the spin density at the position of ligand nuclei and thus gives the important information about the relative magnitude of spindelocalization and spin-polarization effects. Hyperfine interaction results in a shift of NMR resonance frequency and in an enhancement of nuclear spin relaxation. The SBM equations14-17 describe the increase of longitudinal, 1/T1, and transverse, 1/T2, relaxation rates of ligand nuclei in the inner coordination sphere of a paramagnetic complex due to the time-dependent interaction with the electron spin. As in the general case of hyperfine interactions, relaxation rates can also be split into a scalar (Fermi contact) 1/TSC i and a dipolar contribution, 1/TDD i
( )[ ( )[ ( )[
2S(S + 1) A 1 ) SC 3 p T1 S(S + 1) A 1 ) SC 3 p T 2
2
1 + ωS2τs22 τs2
2
1 + ωS2τs22
2 µ0 2 (βeβNgegN) 1 ) S(S + 1) 15 4π TDD rMX6 1
2
τs2
2
(6)
]
where ωS and ωI are the electron and nuclear Larmor frequencies in rad s-1, rMX is the distance between the nucleus of the paramagnetic ion and the ligand nucleus under observation, and τsi and τdi are characteristic correlation times of scalar and dipolar relaxation processes which in turn depend on the correlation times of molecular rotation, τR, on the residence time of the ligand in the inner coordination sphere of the paramagnetic ion, τM, and on electron spin relaxation times, T1e,T2e. Note that A/p in SBM equations is equal to 2πAiso as defined in eq 3. Once the principal values (two independent parameters) of the dipolar HFI tensor T are known (from quantum-chemical calculations, for example), it is possible to overcome the pointdipole approximation and to rewrite the “HFI factor” in the dipolar SBM equations in a generalized form introducing the notation of an “effective” distance reff18-20 for the dipole-dipole interaction,
reff )
(
S βeβNgegN
)[ 2
+ τs1
(7)
]
τd2 13 + 1 + ωS2τd22 τd1 + 4τd1 (9) 3 1 + ωI2τd12 2
]
]
1 Tzz2 + (Txx - Tyy)2 3
-1/6
(10)
where Tzz is the maximal absolute value of the principal components of the dipolar HFI tensor and Txx, Tyy are the other two eigenvalues of T. Because of the 1/r3 dependence of dipolar interactions, even small spin-distribution effects on the ligand nucleus can significantly influence the resulting value of the effective distance of the dipolar hyperfine interaction. For relaxation rates, this dependence is even stronger since HFI enter in square in eqs 6-9 thus resulting in 1/r6 dependence. It is therefore useful to decompose the dipolar HFI into a point-dipole and a ligandcentered contribution:
T ) TPD + TLC
τd2 7 + 1 + ωS2τd22 τd1 (8) 3 1 + ωI2τd12
( )[
µ0 1 1 (βeβNgegN) ) S(S + 1) DD 6 15 4π T2 rMX
]
Figure 1. The structure of the model Gd3+ octaaqua complex of D4d symmetry. The 4-fold symmetry axis is aligned to be vertical.
(11)
Several simplifications have been proposed in the literature for the analysis of ligand-centered contributions.33-35 However, all these models were intrinsically linked to specific chemical systems. The approach presented below is free from specific approximations and still provides a clear insight into relations between the dipolar HFI tensor and TPD. Computational Techniques Model Systems. In this work we present a two-stage computational strategy to calculate dipolar HFI tensors. In the first stage, we studied as model compound [Gd(H2O)8]3+ with a square antiprismatic coordination polyhedron of D4d symmetry (Figure 1). The geometry of the system is the same as described by Borel et al. 36 except for the Gd-O distance which was fixed
11000 J. Phys. Chem. A, Vol. 109, No. 48, 2005 to 2.40 Å for the sake of better agreement with experimental studies37 of Gd3+ in aqueous solution. Due to the small size and the high symmetry of the aquaion, quantum chemical calculations of the complex are rather undemanding. That allowed us to perform a very detailed study of the influence of the basis set quality as well as other computational aspects on the HFI tensor. The second stage involves a study of the [Gd(DOTA)(H2O)]complex in aqueous solution. To take into account solvent effects and to provide sufficient statistical averaging of the calculated HFI tensor we used a cluster method (see Chapter 11 in ref 26 and references therein). This approach implies the ensemble averaging of a property calculated for a set of single configurations (often referred to as “snapshots”) extracted from a molecular dynamics simulation trajectory. This approach has been proven to be reasonable for studying properties such as NMR chemical shifts,38 nuclear quadrupolar coupling constants,39-42 hyperfine coupling constants,43 and g-tensors44 of fluids and biosystems in solution. In our work, we applied this approach to investigate an extended system including almost 100 atoms with the focus on the Gd3+ ion in the complex and on the water molecule directly bound to it. A classical molecular dynamics simulation of the [GdDOTA(H2O)]- complex in water has been performed in a periodic box containing the complex anion, Jorgensen TIP3P water molecules,45 and a Na+ counterion using Amber 6.0 code46 (T ) 300 K, P ) 1 atm, NPT ensemble). The parametrization of [GdDOTA(H2O)]- and the details of the simulation have been described previously.47,48 It is known from experiment that the [GdDOTA(H2O)]- complex is present in solution as a mixture of major M (80%) and minor m (20%) conformers with different coordination polyhedra and ligand conformations.49 Only the major M isomer has been simulated and was therefore considered in the present study. The configuration space sampling (100 snapshots) was extracted from the trajectory at regular intervals of 10 ps. The time series of geometric parameters and calculated properties show no autocorrelation. The single configuration (cluster) for quantum chemical calculations consisted of the [GdDOTA(H2O)]- complex and the 6 second sphere water molecules closest to the inner sphere water molecule (74 atoms in total). Including second sphere water molecules ensures an adequate treatment of close-range solvent effects. Moreover, using the polarizable continuum model (PCM) calculations, we found that far-range solvent effects do not significantly influence the hyperfine coupling constants of the first coordination sphere water molecule. We neglected therefore in our quantum chemical calculations long-range solvent effects, and all calculations were performed for isolated clusters chosen from the snapshots. A typical example of a single configuration is shown in Figure 2. Theoretical Methods. Calculations of lanthanide compounds as well as those of other heavy elements require an adequate treatment of relativistic effects.50 In general, there are a few alternative ways to treat relativistic effects. Usually one has to chose between all-electron treatment (including only scalar or both scalar and spin-orbit relativistic effects) and relativistic effective core potentials (RECP), which themselves can also be pure scalar or include as well the spin-orbital part. The use of RECP can significantly reduce computational efforts since core electrons are removed and replaced by an effective operator. Several RECP parametrizations for gadolinium51,52 are available for routine applications. These pseudopotentials were proven to reproduce reliably experimental molecular geometries and vibrational spectra of gadolinium compounds.53,54 However, we
Yazyev et al.
Figure 2. A typical “snapshot” extracted from MD simulation of [GdDOTA(H2O)]-. The Gd3+ ion and first coordination sphere water molecule are shown in balls and sticks; the polyaminocarboxylate ligand DOTA and 6 second coordination sphere water molecules are presented as tubes.
found that for HFI tensor calculations all-electron treatments are much better than RECP approaches we considered.55 Therefore, in the following we will only discuss the results of all-electron approaches. Currently, among the all-electron relativistic approaches the family of Douglas-Kroll-Hess (DKH) transformation56 based methods and zero order regular approximation (ZORA)57-59 are the methods most widely tested and used. In a first stage of our computational study we compare second order Douglas-KrollHess method (DKH2) implemented in the Gaussian03 suite of programs60 with the ZORA method available in ADF2003 package.61 For consistency the point-nucleus model was used in both methods (this is a good approximation in the present study since we are interested in HFI tensors on ligand nuclei only). Spin-orbit coupling terms were not taken into account since they were not expected to be important for HFI tensor calculations due to the electronic structure and the high coordination (8 or 9) of Gd3+ in the compounds under study. In addition, recent calculations of spin-orbit corrections to HFI tensor for lighter nuclei found only minor effects.62,63 The choice of basis set becomes a nontrivial task if one considers calculations of isotropic hyperfine coupling constants. There are two main problems associated with basis sets in such calculations. First, the accurate representation of nuclear cusp is necessary for evaluation of spin density at the point of nucleus. While this can be naturally covered using Slater type orbital (STO) basis sets, extra tight exponents are needed for Gaussian type orbital (GTO) basis sets. Second, it is not feasible to use large contractions (in the case of GTO basis sets) for the description of the core region because the additional flexibility is required to take into account spin-polarization effects.64,65 Therefore, the core basis functions have to be considerably uncontracted if GTO basis sets are used or simply represented by a sufficient number of functions of STO basis sets. The frozen core approximation is unacceptable in calculations of HFI if the core MOs are frozen on the nucleus of interest. In the present study, we used both GTO and STO basis sets for the calculations of the model Gd3+ octaaqua complex. The GTO set used in all DKH2 calculations was composed from relativistic basis sets (Hirao et al.66) by complete uncontraction and, in addition, for the light atoms, it was augmented with the polarization functions from the IGLO-III basis set.67 In ZORA calculations on [Gd(H2O)8]3+ we used the standard TZ2P STO basis set from the ADF package. However, here the concept of basis set “of
Hyperfine Interactions in Gd(III) Complexes high local quality” 68 (locally dense basis set) was used in order to reduce the amount of computational resources required. In our case the TZ2P basis set was used only for the part of interest, Gd3+ and first coordination sphere water molecules, while the rest of the system was treated using the DZ basis set with frozen 1s core. On Gd the frozen core up to the 4d shell was used while no frozen core approximation was employed for the atoms of first coordination sphere water molecules. Using model calculations we showed that this approximation does not affect superhyperfine coupling constants on the ligand nuclei. While most quantum chemical calculations are nowadays performed with the DFT approach, the question about its performance in calculations of a particular property is not an easy one. DFT for calculations of HFI of organic radicals gives acceptable results, and good accuracy can be achieved with pure density functionals except for some pathological cases. However, transition metal complexes are much more difficult to calculate, and some pictorial failures of pure density functionals are known.69 For instance, DFT tends to overestimate the coordination bond covalency in Cu2+, which in turn leads to a more delocalized character of SOMO in such systems. Moreover, there is no a priori known best exchange-correlation functional for calculation of hyperfine coupling constants: all GGA functionals behave more or less similarly while hybrid density functionals usually give better results.64,69 The use of hybrid density functionals (especially with a large admixture of HF exchange) increases, however, spin contamination, which could lead to inferior results.64 The electronic structure of lanthanide compounds differs significantly from d-transition metal complexes: the core character of unpaired f-shell electrons makes an admixture of excited states energetically unfavorable. Thus, in principle, an admixture of the Hartree-Fock exchange should not lead to severe spin-contamination. The question about the performance of DFT for the description of spin-polarization driven effects on HFI tensor in lanthanides is still unexplored, and we present here a first attempt of such benchmark calculations. Among available exchange-correlation density functionals the exchange functional of Becke70 and the correlation functional of Perdew and Wang71 (this combination is known as the BPW91 functional) were chosen relying on benchmark calculations from Kaupp et al.64 Thus we used BPW91 as a pure DFT functional, B3PW9172 as its hybrid modification, and the Hartree-Fock method for completeness of the consideration. All methods were used in their spinunrestricted implementation necessary to take into account spinpolarization effects. In all calculations performed with the Gaussian03 package the tight SCF convergence criterion (SCF)tight keyword) and fine numerical integration grids (Integral(FineGrid) keyword) have been used. The use of symmetry in the evaluation and storage of integrals was disabled. In ADF2003 calculations the numerical integration parameter of 6.0 was used (Integration 6.0 keyword). We proved that using higher convergence criteria and more accurate integration grids does not influence values of calculated HFCCs. The calculations were performed on a homemade PC cluster. Results and Discussion Validation of Methodology of Quantum Chemical Calculations. The detailed calculations on the small model [Gd(H2O)8]3+ allowed us to assess the reliability of density functional theory calculations of HFI for Gd complexes. The obtained isotropic 17O hyperfine coupling constants (HFCC) (Table 1) are between 0.61 and 1.02 MHz for the DFT
J. Phys. Chem. A, Vol. 109, No. 48, 2005 11001 TABLE 1: Calculated 17O and 1H Hyperfine Tensors for the First Coordination Sphere Water Molecules of the [Gd(H2O)8]3+ Model Systema method
Aiso, MHz
Txx, MHz
Tyy, MHz
0.802 0.761 0.716 0.681 0.775
0.560 0.583 0.633 0.671 0.775
-1.362 -1.344 -1.349 -1.352 -1.551
-2.652 -2.651 -2.655 -2.653 -2.666 -2.663 -2.677 -2.674 -2.643
-2.625 -2.626 -2.623 -2.623 -2.618 -2.618 -2.606 -2.607 -2.643
5.277 5.277 5.278 5.275 5.283 5.281 5.283 5.280 5.287
17O
BPW91/ZORA/TZ2P BPW91/DKH2/Hirao B3PW91/DKH2/Hirao HF/DKH2/Hirao exptl/point-dipoleb
0.61 0.78 1.02 1.60 0.84
1H 0.022 0.028 BPW91/DKH2/Hirao -0.001 0.009 B3PW91/DKH2/Hirao -0.031 -0.023 HF/DKH2/Hirao -0.044 -0.029 exptl/point-dipoleb 0.03 ( 0.02
BPW91/ZORA/TZ2P
Tzz, MHz
reff, Å 2.50 2.51 2.51 2.51 2.40 3.105 3.106 3.105 3.106 3.104 3.105 3.104 3.105 3.1037
a Two different values for 1H correspond to two types of protons a in the model system. H1 and H2 protons as shown in Figure 1 belong to the same water molecule and are equivalent for different water molecules. H1 protons are equatorial and H2 are axial with respect to the symmetry axis. The Gd-H1 and Gd-H2 distances are equal. b reff obtained using the point-dipole approximation.
Figure 3. Spin density map of the [Gd(H2O)8]3+ model system (calculated at the BPW91/DKH2/Hirao level of theory, in au-3) shows the Gd3+ ion and one of the water molecules. Cross sections of size 2 × 4 Å in XOZ and YOZ planes are shown.
calculations. An admixture of the HF exchange to the exchangecorrelation potential (B3PW91 vs BPW91) pushes the calculated 17O HFCC in the direction of the HF results, which is not unexpected. The difference between ZORA and DKH results for isotropic 17O HFCC can be attributed to the neglect of socalled picture-change effect in present DKH calculations.73 Taking into account that, due to a very local character of the Fermi contact operator, the isotropic constants are very difficult to evaluate computationally, the results show reasonable mutual agreement. Furthermore, the DFT calculations are consistent with the experimental values of 0.84 MHz74,75 and 0.71 MHz76 for [Gd(H2O)8]3+. Our model calculations do not include longrange solvent effects which are expected to be insignificant for HFI. Therefore, both pure and hybrid density functionals look reliable to describe ligand nuclei HFI in the compounds studied. The positive isotropic HFCC, Aiso, on 17O nuclei corresponds to a negative spin-density at the point of the O nucleus of -0.0118 au-3 while the most negative value of spin-density within cross sections shown in Figure 3 is -0.1342 au-3. The calculated Tzz values (Table 1) of the 17O HFI anisotropy tensor are noticeably lower (about -1.35 MHz) than predicted by the point-dipole approximation (-1.55 MHz) using eq 5 and rGd-O ) 2.40 Å. All quantum chemical methods used in this study give results in good mutual agreement, which is not surprising since it is well-known that HFI anisotropy is much
11002 J. Phys. Chem. A, Vol. 109, No. 48, 2005 less sensitive to the computational aspects than the isotropic part.26 Again this excellent agreement supports our confidence in using the DFT method in the calculations of HFI on ligand nuclei in lanthanide compounds. The rhombicity of HFI anisotropy tensor, Txx - Tyy, depends on the choice of quantum chemical method. The pure density functional BPW91 yields the largest positive Txx - Tyy values (0.242 and 0.178 MHz) as a consequence of possible residual π-binding involving felectrons. This is not surprising since pure density functionals are known to overestimate the covalency of coordination bond. However, for Gd3+ the rhombicity influences only slightly the effective distance reff (eq 10), which is almost solely defined by the Tzz value. Our calculations of the HFI anisotropy tensor are in qualitative agreement with the estimations of Raitsimring et al.77 In their calculations they estimate the spin population of the whole s-pz hybrid orbital of the O atom from the experimental isotropic 17O HFCC and neglect any valence shell and core shell spin-polarization effects. We have shown recently that these effects might be significant and can be taken into account within quantum chemical calculations.78 The calculated 1H isotropic hyperfine couplings are very small (Table 1). The increase of amount of exact exchange tends to decrease spin density at 1H nucleus and even to change its sign. This can be attributed to the fact that the hydrogen atoms lie very close to a node of the spin-density surface (see Figure 3). The calculated 1H HFI anisotropy tensors are in very good agreement with each other and with the point-dipole approximation. All tested quantum chemistry methods give a negligible ligand-centered contribution to the 1H HFI tensor. The basis sets used in our calculations are sufficient since calculated HFCC’s do not show significant changes upon the addition of extra basis functions. The purity of spin states is proven by calculated 〈S2〉 values (for DFT methods we use Kohn-Sham determinant to evaluate 〈S2〉). For BPW91, B3PW91, and HF calculations with GTO basis set the calculated values of 〈S2〉 are 15.7556, 15.7556, and 15.7576, correspondingly, while the nominal value for a pure octet state is 15.75. Hyperfine Coupling Constants of 17O and 1H in [Gd(H2O)8]3+. The qualitative inspection of spin density maps presented graphically in Figure 3 shows the strong spinpolarization effect on the water molecule. While most of the positive spin density resides on the Gd3+ ion itself, electron density along Gd-O bond is significantly spin-polarized, leading to two important consequences. First, the calculated Tzz value (OZ axis is oriented along Gd-O bond) of -1.35 MHz for 17O (Table 1) is noticeably lower than the value of -1.55 MHz predicted by the pointdipole approximation (eq 5). This reduction of the Tzz value leads to an effective distance of reff ) 2.50 Å for the dipoledipole interaction (eq 10), which is considerably longer than the Gd-O internuclear distance rGd-O ) 2.40 Å (Table 2). One can think about this effect as a partial compensation of magnetic dipole-dipole interaction between the positive spin-density on the Gd3+ ion and the 17O nucleus by the negative spin-density induced on the ligand. Furthermore, it is interesting to note the positive spin density located in the YOZ plane perpendicular to the plane of the water molecule (Figure 3). This is reflected in the bigger value of Txx - Tyy for BPW91 calculations and can be a consequence of some residual π-binding involving f-electrons or a spin alternation effect. However, as it was discussed above, the small rhombicity of the oxygen HFI anisotropy tensor almost does not influence the resulting value of the effective distance of dipolar interaction reff(Gd-O).
Yazyev et al. TABLE 2: Comparison of Calculated (BPW91/ZORA/ TZ2P) and Experimental Hyperfine Interaction Parameters for [Gd(H2O)8]3+ and [Gd(L)(H2O)]n- a [Gd(H2O)8]3+ calcd
[Gd(L)(H2O)]n-
exptl
calcdb
exptl
17O
rGd-O, Åc reff, Åd Aiso, MHz
2.4 2.50 0.61
Txx, MHz Tyy, MHz Tzz, MHz
0.802 0.560 -1.362
0.79e 0.84f 0.75g 0.76g,l 0.62g,l -1.38g
2.56 (0.06) 2.72 (0.06) 0.58 (0.11)
0.59h 0.75i
0.623 0.452 -1.061 (0.09)
0.76i 0.62i -1.38i
3.27 (0.14) 3.27 (0.14) -0.032 (0.08)
3.06i -0.04k
-2.306 -2.256 4.562 (0.66)
-2.75k -2.75k 5.5k
1H
rGd-H, Å reff, Å Aiso, MHz Txx, MHz Tyy, MHz Tzz, MHz
3.1037 3.106 0.025 -2.652 -2.626 5.277
3.09j 0.04e 0.03j -2.67j -2.67j 5.34j
a The distribution widths from molecular dynamics sampling are given in parentheses. b L ) DOTA. c Nuclear distance. d Effective distance of dipole-dipole interaction. e From NMR chemical shift (ref 76), corrected for coordination number of 8. f From NMR chemical shift (ref 75). g From ENDOR experiments (ref 80). h L ) DOTA, NMR chemical shift (ref 75). i L ) MS-325, ENDOR experiments (ref 77). j From ENDOR (ref 80). k L ) HPDO3A, ENDOR experiments (ref 80). l From the estimated rhombicity of 0.14 MHz for the planar model 2 (ref 77).
Second, the 17O isotropic hyperfine coupling constant, Aiso, is positive as the consequence of a negative spin density at the oxygen nucleus and the negative magnetic moment of the 17O nucleus. As we already mentioned in the beginning of this section, the experimental values of 0.71 MHz76 and 0.84 MHz74 lie within the 0.61-1.02 MHz range of DFT predictions (Table 1).79 The magnitude of the coupling is relatively small since the s-pz hybrid atomic orbital of O, which is mostly affected by spin-polarization, has little of s-character and therefore its node lies very close to the nucleus. The Gd-H effective distance of dipole-dipole magnetic interaction recalculated via data of quantum chemical calculations (Table 2) is only slightly bigger than the distance between the nuclei, rGd-H. The point-dipole approximation is therefore valid for hydrogens of water molecules in [Gd(H2O)8]3+. The validity of this approximation for 1H has already been found in a previous study on dipolar HFI in d-transition metal aquaions.19 The rationalization of this observation is that p-type atomic orbitals on hydrogen play only a minor role in bonding (and thus could not contribute significantly to the anisotropy of the HFI tensor) whereas the s-type atomic orbital gives zero contribution to the anisotropy of the tensor (but determines the isotropic constant Aiso). The calculated 1H isotropic hyperfine couplings are close to zero and in good agreement with experimental data. For instance, single-crystal EPR studies80 put this value between -0.015 and +0.04 MHz while the most reliable ENDOR81 gives +0.03 ( 0.02 MHz. Bryden et al.82 deduced 1H HFCC from NMR data to be about 0.005 MHz. We conclude that there is rather large uncertainty in the experimental data due to the small absolute magnitude of the coupling. Isotropic Hyperfine Coupling Constants of 17O and 1H in [Gd(DOTA)(H2O)]-. The isotropic 17O HFCCs obtained using the DFT cluster approach with averaging over a set of snapshots selected from a classical molecular dynamic simulation are in
Hyperfine Interactions in Gd(III) Complexes
Figure 4. 17O isotropic hyperfine coupling constant, Aiso, plotted as function of Gd-O distance for 100 configurations extracted from MD trajectory of [Gd(DOTA)(H2O)]-.
Figure 5. 1H isotropic hyperfine coupling constant plotted as function of Gd-H distance for 100 configurations (200 values totally) extracted from MD trajectory.
a very good agreement with experimental data. A major parameter influencing the 17O HFCC is the Gd-O distance, rGd-O, which fluctuates during the molecular dynamics simulation. Figure 4 shows the isotropic hyperfine coupling constant as a function of rGd-O. The averaged Aiso(O) is 0.58 MHz (standard deviation: 0.11 MHz) for an average distance of 2.56 ( 0.06 Å; that is in excellent agreement with the value of 0.59 ( 0.03 MHz determined from 17O NMR chemical shift data.75 In all geometric configurations obtained from the snapshots the spin density at the position of oxygen nucleus is negative in accord with the spin polarization mechanism. The magnitude of Aiso(O) strongly correlates with the Gd-O distance and the spin-polarization effect decays rapidly with the increase of rGd-O. The mean Gd-O distance from the classical MD simulation is however about 0.1 Å longer than that of the solid-state X-ray83 or the solution XAFS structure.84 17O electron nuclear double resonance spectra of the MS-325, a Gd3+ complex with an acyclic ligand, recorded in frozen solution, gave spectra of a shape similar to that of the Gd3+ aquaion.77 The authors concluded therefore that the 17O hyperfine coupling parameters of both complexes are the same, Aiso ) 0.75 MHz. The calculated mean value of the 1H isotropic hyperfine coupling constant (-0.032 ( 0.08 MHz) is very small and varies from about -0.2 to + 0.1 MHz (Figure 5). Again a strong correlation with the Gd-H distance is observed. The corresponding NMR experimental value of Bryden et al.82 for [Gd(DOTA)(H2O)]- is about 0.075 MHz while the twodimensional Mims ENDOR result81 for [Gd(HP-DO3A)(H2O)]-
J. Phys. Chem. A, Vol. 109, No. 48, 2005 11003
Figure 6. The reff-6(Gd-O) factor for the oxygen atom of bound water in [GdDOTA(H2O)]- obtained from eq 10 based on the calculated dipolar HFI tensors of the 100 MD “snapshots”. The line corresponds to r-6(Gd-O) calculated using the point-dipole approximation.
Figure 7. The reff-6(Gd-H) factor for the hydrogen atom of bound water in [GdDOTA(H2O)]- obtained from eq 10 based on the calculated dipolar HFI tensors of the 100 MD “snapshots” (200 values totally). The line corresponds to r-6(Gd-H) calculated using the point-dipole approximation.
is -0.04 ( 0.02 MHz. Regarding the slightly longer Gd-H average distance from the MD simulations (rGd-H ) 3.27 Å compared to 3.1 Å from ENDOR) the agreement between calculated and experimental Aiso(H) is also very satisfactory. Anisotropic Hyperfine Coupling Constants of 17O and 1H in [Gd(DOTA)(H2O)]-. The results of anisotropic HFCCs are expressed in terms of effective distances of dipole-dipole interactions, reff (eq 5), for the reader’s benefit and better understanding. It will furthermore allow us to compare the results of the calculations with those obtained within the pointdipole approximation. Figures 6 and 7 show the values of the reff-6 factor as function of the Gd-ligand nucleus distances. The values of reff-6, obtained from eq 10, are based on the calculated dipolar HFI tensors for 100 MD snapshots. The pointdipole approximation used in the SBM equations uses internuclear distances rGd-X, and the corresponding factors rGd-X-6 are shown as solid lines in Figures 6 and 7. From Figure 6 it can be seen that for 17O the point-dipole approximation significantly overestimates the dipolar interaction between the ligand nuclear spin and the electron spin of the ion: The expectation value of the effective distance factor for dipolar relaxation is 〈reff-6(Gd-O)〉 ) 2.51 × 10-3 Å-6 compared to the internuclear gadolinium-oxygen distance of 〈r-6(Gd-O)〉 ) 3.64 × 10-3 Å-6. This corresponds to the following average distances of dipole-dipole magnetic interac-
11004 J. Phys. Chem. A, Vol. 109, No. 48, 2005 tions: 〈reff(Gd-O)〉 ) 2.72 Å and 〈r(Gd-O)〉 ) 2.56 Å, respectively. Neglecting spin-polarization effects leads therefore to an overestimation of the dipolar 17O nuclear spin relaxation rates by approximately 45%. The point-dipole approximation is however valid to describe the anisotropic HFI between Gd3+ and 1H of the inner sphere water molecule in [Gd(DOTA)(H2O)]-. From Figure 7 one can note that reff-6(Gd-H) deviates only very slightly from r-6(Gd-H) at short Gd-H distances and the average values are virtually identical: 〈reff-6(Gd-H)〉 ) 8.48 × 10-4 Å-6 and 〈r-6(Gd-H)〉 ) 8.56 × 10-4 Å-6. Conclusions The study of physicochemical parameters of gadolinium based MRI contrast agents is to a large extent based on 1H and 17O NMR relaxation measurements. The strong interaction between the nuclear spins and the electron spin of Gd3+ dominates the relaxation properties and allows the determination of rotational correlation times and water exchange rates, for example. Thus, an accurate knowledge of the isotropic and anisotropic parts of the hyperfine coupling constant is imperative for an accurate assessment of the data. Our quantum chemical investigation of hyperfine coupling constants between Gd3+ ion and bound water molecules shows that the 1H isotropic hyperfine coupling constants, Aiso, are small. Thus, the contribution of scalar relaxation to the relaxation enhancement can therefore be safely neglected as it has already been deduced from experimental data. What is more essential, the evaluation of the anisotropic part of the A-tensor shows that the point-dipole approximation is valid for 1H NMR of bound water molecules. The distance rGd-H introduced in SBM theory can therefore be safely set to the distance between the gadolinium and the proton nuclei. Reduced transverse 17O NMR relaxation rates 1/T2r are, in the case of Gd-based complexes, dominated by the scalar relaxation mechanism which depends on Aiso. This isotropic hyperfine coupling constant can be best determined experimentally by 17O NMR shift measurements. Our calculated Aiso(O) strongly correlates with the metal-oxygen distance, and therefore, NMR shift measurements give at least a qualitative indication for rGd-O: smaller coupling constants are indicative for longer Gd-O distances. Reduced longitudinal 17O NMR relaxation rates, 1/T1r, are controlled to about two-thirds by the dipolar contribution, 1/TDD 1 , and to about one-third by the quadrupolar contribution, 1/TQ1 .3 The quantum chemical evaluation of the anisotropic HFI tensor T has shown that spin-polarization effects play an essential role for proper evaluation of 17O HFI tensor. The pointdipole approximation, based on internuclear distances, significantly overestimates the interaction between electron and 17Onuclear spin. To compensate for the 45% decrease of 1/TDD 1 by using the correct distance for dipolar interaction (2.72 Å instead of 2.46 Å) a ∼25% increase of the rotational correlation time τR is needed. Acknowledgment. We thank Dr. Alain Borel for helpful discussions. The Swiss National Science Foundation is acknowledged for financial support. Work in Bratislava has been supported by the Slovak Grant Agencies VEGA (No. 2/3103/ 23) APVT (No. 51-045502), EURATOM FP6 Integrated Project (FUNMIG), and European COST D18 action “Lanthanide Chemistry for Diagnosis and Therapy”. This research was carried out in the frame of the European-funded EMIL program (LSHC -2004 - 503569).
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